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Properties

Label 6-165e3-1.1-c5e3-0-3
Degree $6$
Conductor $4492125$
Sign $-1$
Analytic cond. $18532.4$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 27·3-s − 32·4-s − 75·5-s − 54·6-s − 232·7-s + 96·8-s + 486·9-s + 150·10-s + 363·11-s − 864·12-s + 450·13-s + 464·14-s − 2.02e3·15-s − 112·16-s − 334·17-s − 972·18-s − 4.03e3·19-s + 2.40e3·20-s − 6.26e3·21-s − 726·22-s − 7.06e3·23-s + 2.59e3·24-s + 3.75e3·25-s − 900·26-s + 7.29e3·27-s + 7.42e3·28-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.73·3-s − 4-s − 1.34·5-s − 0.612·6-s − 1.78·7-s + 0.530·8-s + 2·9-s + 0.474·10-s + 0.904·11-s − 1.73·12-s + 0.738·13-s + 0.632·14-s − 2.32·15-s − 0.109·16-s − 0.280·17-s − 0.707·18-s − 2.56·19-s + 1.34·20-s − 3.09·21-s − 0.319·22-s − 2.78·23-s + 0.918·24-s + 6/5·25-s − 0.261·26-s + 1.92·27-s + 1.78·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(4492125\)    =    \(3^{3} \cdot 5^{3} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(18532.4\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 4492125,\ (\ :5/2, 5/2, 5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p^{2} T )^{3} \)
5$C_1$ \( ( 1 + p^{2} T )^{3} \)
11$C_1$ \( ( 1 - p^{2} T )^{3} \)
good2$S_4\times C_2$ \( 1 + p T + 9 p^{2} T^{2} + 5 p^{3} T^{3} + 9 p^{7} T^{4} + p^{11} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 + 232 T + 48289 T^{2} + 7415840 T^{3} + 48289 p^{5} T^{4} + 232 p^{10} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 - 450 T + 1068439 T^{2} - 311504212 T^{3} + 1068439 p^{5} T^{4} - 450 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 + 334 T + 3997931 T^{2} + 890682028 T^{3} + 3997931 p^{5} T^{4} + 334 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 + 4036 T + 10546833 T^{2} + 18356119928 T^{3} + 10546833 p^{5} T^{4} + 4036 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 + 7060 T + 25138261 T^{2} + 66605461592 T^{3} + 25138261 p^{5} T^{4} + 7060 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 - 4042 T + 41491883 T^{2} - 99862913932 T^{3} + 41491883 p^{5} T^{4} - 4042 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 + 608 T - 4957539 T^{2} - 176765401280 T^{3} - 4957539 p^{5} T^{4} + 608 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 - 2250 T + 76046251 T^{2} - 743927675036 T^{3} + 76046251 p^{5} T^{4} - 2250 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 - 10654 T + 367707807 T^{2} - 2476467488116 T^{3} + 367707807 p^{5} T^{4} - 10654 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 + 35528 T + 861670557 T^{2} + 12105543975808 T^{3} + 861670557 p^{5} T^{4} + 35528 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 + 2100 T + 593888701 T^{2} + 1015411414488 T^{3} + 593888701 p^{5} T^{4} + 2100 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 + 242 p T + 419938891 T^{2} + 13415490019292 T^{3} + 419938891 p^{5} T^{4} + 242 p^{11} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 + 81876 T + 3647590801 T^{2} + 113436530018552 T^{3} + 3647590801 p^{5} T^{4} + 81876 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 + 62298 T + 3103700755 T^{2} + 92329281973340 T^{3} + 3103700755 p^{5} T^{4} + 62298 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 + 46148 T + 3289111289 T^{2} + 83962738468760 T^{3} + 3289111289 p^{5} T^{4} + 46148 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 + 64724 T + 6737647765 T^{2} + 242132017763608 T^{3} + 6737647765 p^{5} T^{4} + 64724 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 - 810 T + 973289755 T^{2} - 147790502790292 T^{3} + 973289755 p^{5} T^{4} - 810 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 - 43876 T - 340410235 T^{2} + 81866739123944 T^{3} - 340410235 p^{5} T^{4} - 43876 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 + 101024 T + 13571488893 T^{2} + 801669575220272 T^{3} + 13571488893 p^{5} T^{4} + 101024 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 - 60022 T + 6543211047 T^{2} - 916436193435188 T^{3} + 6543211047 p^{5} T^{4} - 60022 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 + 319746 T + 51760850431 T^{2} + 5671445258599612 T^{3} + 51760850431 p^{5} T^{4} + 319746 p^{10} T^{5} + p^{15} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.11166507778935650298849369025, −10.45728332715078813167465920634, −10.26432736657659890824356406804, −10.12212587692076421456333593473, −9.390463505813236589480313996394, −9.276701086529893061591256878817, −9.196263354115739681207289537677, −8.464866503631378434479604307829, −8.382083376923113653343472722928, −8.334723813427079207734442182447, −7.63361200949124834980424528031, −7.55467052155738386203294011291, −6.65570257910159012740639350911, −6.62754375736756506334974633803, −6.15650676064698743725429505115, −6.01252342373872770637520303925, −4.62418724297109671954963949375, −4.45326433144168677446687328834, −4.36323270705408538979636450099, −3.64463929107714971973204415576, −3.60938486706975797715607150314, −3.04182088284471260804033926316, −2.56966967452185228844934593381, −1.68659459446184327513572108682, −1.47606669972464682093444461920, 0, 0, 0, 1.47606669972464682093444461920, 1.68659459446184327513572108682, 2.56966967452185228844934593381, 3.04182088284471260804033926316, 3.60938486706975797715607150314, 3.64463929107714971973204415576, 4.36323270705408538979636450099, 4.45326433144168677446687328834, 4.62418724297109671954963949375, 6.01252342373872770637520303925, 6.15650676064698743725429505115, 6.62754375736756506334974633803, 6.65570257910159012740639350911, 7.55467052155738386203294011291, 7.63361200949124834980424528031, 8.334723813427079207734442182447, 8.382083376923113653343472722928, 8.464866503631378434479604307829, 9.196263354115739681207289537677, 9.276701086529893061591256878817, 9.390463505813236589480313996394, 10.12212587692076421456333593473, 10.26432736657659890824356406804, 10.45728332715078813167465920634, 11.11166507778935650298849369025

Graph of the $Z$-function along the critical line